Why a part of my mesh disappears and its untexturable?
I just played about with a much more basic mesh, mirrored in X &amp; Y like yours, and it looks to me like the rest of the mesh doesn't actually 'exist' for UV mapping etc. until you've applied the modifier.
Safest bet is to duplicate the object (select then hit Shift+D in 'object' mode) and then apply the mirror mod on the new object, unwrap &amp; texture that.
Then if you want to change the model, edit the original mirror-modified object, and duplicate &amp; apply the mirror modifier again.

Here's my  test- 

So the only geometry that truly 'exists' is the bit that turns orange in Edit mode, until you apply the mirror modifier, all you'll get is the same UV'd texture mirrored.
If you unwrap your model before applying the modifier, you should only see the 'real' geometry in the UV editor, so that's all the UV'd image or whatever will be applied to (the rest of the model will get the same texture but mirrored the same as the geometry).
Like this- 
As you can see there's only the 'real' quater of the mirrored object in the UVs when unwrapped.

Hope this helps...